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Let's solve the two questions one by one.

**(a) Finding the average rate of change from 0 minutes to 20 minutes**

The average rate of change of temperature \( R(t) \) over a time interval \([t_1, t_2]\) is calculated by:
\[
\text{Average rate of change} = \frac{R(t_2) - R(t_1)}{t_2 - t_1}
\]

For the interval from 0 to 20 minutes:
- \( t_1 = 0 \) and \( R(t_1) = 184.3 \)
- \( t_2 = 20 \) and \( R(t_2) = 140.3 \)

Substitute these values:
\[
\text{Average rate of change} = \frac{140.3 - 184.3}{20 - 0} = \frac{-44}{20} = -2.2 \, \text{°C per minute}
\]

So, the answer for part (a) is **-2.2 °C per minute**.

---

**(b) Finding the average rate of change from 30 minutes to 60 minutes**

For the interval from 30 to 60 minutes:
- \( t_1 = 30 \) and \( R(t_1) = 109.3 \)
- \( t_2 = 60 \) and \( R(t_2) = 28.3 \)

Substitute these values:
\[
\text{Average rate of change} = \frac{28.3 - 109.3}{60 - 30} = \frac{-81}{30} = -2.7 \, \text{°C per minute}
\]

So, the answer for part (b) is **-2.7 °C per minute**.

Would you like a detailed breakdown or have any questions about this? Here are some related questions:

1. What is the significance of a negative rate of change in temperature?
2. How would the average rate of change differ if calculated over a longer time interval?
3. How can the rate of temperature decrease be used to predict cooling time?
4. What mathematical model might describe the cooling of a roast?
5. How would you find the instantaneous rate of change of temperature at a specific time?

**Tip:** The average rate of change gives a general trend, while the instantaneous rate of change gives precise information at a particular moment.

Find the average rate of change for the temperature from 0 to 20 minutes and from 30 to 60 minutes.

Learn how to calculate the average rate of change for the temperature of a roast from 0 to 20 minutes and from 30 to 60 minutes. This problem demonstrates key concepts in rate of change and uses the formula (R(t2) - R(t1)) / (t2 - t1) to find the cooling rate in degrees Celsius per minute.

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